# rays(NormalToricVariety) -- get the rays of the associated fan

## Synopsis

• Usage:
rays X
• Function: rays
• Inputs:
• Outputs:
• a list, of lists of integers; each entry corresponds to a minimal nonzero lattice point on the ray in the underlying fan

## Description

A normal toric variety corresponds to a strongly convex rational polyhedral fan in affine space. In this package, the fan associated to a normal d-dimensional toric variety lies in the rational vector space d with underlying lattice N = ℤd. As a result, each ray in the fan is determined by the minimal nonzero lattice point it contains. Each such lattice point is given as a list of d integers.

The examples show the rays for the projective plane, projective 3-space, a Hirzebruch surface, and a weighted projective space. There is a canonical bijection between the rays and torus-invariant Weil divisor on the toric variety.

 `i1 : PP2 = toricProjectiveSpace 2;` ```i2 : rays PP2 o2 = {{-1, -1}, {1, 0}, {0, 1}} o2 : List``` ```i3 : dim PP2 o3 = 2``` ```i4 : weilDivisorGroup PP2 3 o4 = ZZ o4 : ZZ-module, free``` ```i5 : PP2_0 o5 = PP2 0 o5 : ToricDivisor on normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))```
 `i6 : PP3 = toricProjectiveSpace 3;` ```i7 : rays PP3 o7 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}} o7 : List``` ```i8 : dim PP3 o8 = 3``` ```i9 : weilDivisorGroup PP3 4 o9 = ZZ o9 : ZZ-module, free```
 `i10 : FF7 = hirzebruchSurface 7;` ```i11 : rays FF7 o11 = {{1, 0}, {0, 1}, {-1, 7}, {0, -1}} o11 : List``` ```i12 : dim FF7 o12 = 2``` ```i13 : weilDivisorGroup FF7 4 o13 = ZZ o13 : ZZ-module, free```
 `i14 : X = weightedProjectiveSpace {1,2,3};` ```i15 : rays X o15 = {{-2, -3}, {1, 0}, {0, 1}} o15 : List``` ```i16 : weilDivisorGroup X 3 o16 = ZZ o16 : ZZ-module, free```

When the normal toric variety is nondegerenate, the number of rays equals the number of variables in the total coordinate ring.

 ```i17 : #rays X == numgens ring X o17 = true```

In this package, an ordered list of the minimal nonzero lattice points on the rays in the fan is part of the defining data of a toric variety.